Sur quelques extensions au cadre Banachique de la notion d'op\'erateur de Hilbert-Schmidt
Said Amana Abdillah, Jean Esterle (IMB), Bernhard Hermann Haak (IMB)

TL;DR
This paper explores various extensions of Hilbert-Schmidt operators to Banach spaces, introducing new classes like pre-Hilbert-Schmidt operators and analyzing their properties and relationships.
Contribution
It defines the class of pre-Hilbert-Schmidt operators in Banach spaces and investigates their properties, extending the classical Hilbert-Schmidt concept beyond Hilbert spaces.
Findings
Introduction of the class PS_2(E; F) of pre-Hilbert-Schmidt operators.
Characterization of these operators in the context of Banach spaces.
Connection to existing classes like p-summing and γ-radonifying operators.
Abstract
In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators: -summing operators, -summing or -radonifying operators, weakly -nuclear operators and classes of operators defined via factorization properties. We introduce the class of pre-Hilbert-Schmidt operators as the class of all operators such that is Hilbert-Schmidt for every bounded operator and every bounded operator , where et are Hilbert spaces. Besides the trivial case where one of the spaces or is a "Hilbert-Schmidt space", this space seems to have been described only in the easy situation where one of the spaces or is a Hilbert space.
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Taxonomy
TopicsAdvanced Banach Space Theory · Holomorphic and Operator Theory · Advanced Operator Algebra Research
